Answer:
Figure g: x = 52° , y = 19°
Figure h: a = 132° , b = 28°
Step-by-step explanation:
Figure g:
The measure of the pair of linear angles is 180°
The angle of measure 142°, and its adjacent angle which is inscribed angle in the circle are formed a linear pair of angles
∴ The measure of the inscribed angle + 142 = 180
- Subtract 142 from both sides
∴ The measure of the inscribed angle = 38°
The measure of the central angle is twice the measure of the inscribed angle which subtended by the same arc
∴ The measure of the central angel which subtended by the
same arc with the inscribed angle of measure 38 = 2 × 38
∴ The measure of the central angel = 76°
∵ All the radii in the circle are equal
∵ In Δ which contains the central angle of measure 76 and x
two side of it are radii
∴ The Δ is an isosceles triangle
∴ The measures of its base angles are x° , x°
∵ The sum of measures of their angles is 180°
∴ x + x + 76 = 180
- Add like terms in the left hand side
∴ 2 x + 76 = 180
- Subtract 76 from both sides
∴ 2 x = 104
- Divide both sides by 2
∴ x = 52
In the Δ whose sides are chords in the circle (large triangle)
∵ This Δ contains angle of measures 38°, (x + y)° , (x + 19)°
∵ x = 52
∵ The sum of the measures of the angles of a Δ is 180°
∴ 38 + (52 + y) + (52 + 19) = 180
- Add the like terms in the left hand side
∴ 161 + y = 180
- Subtract 161 from both sides
∴ y = 19°
Figure h:
The measure of the pair of linear angles is 180°
The angle of measure 132°, and its adjacent angle which is inscribed angle in the circle are formed a linear pair of angles
∴ The measure of the inscribed angle + 132 = 180
- Subtract 132 from both sides
∴ The measure of the inscribed angle = 48°
In any inscribed quadrilateral in the circle the sum of the measure of two opposite angle is 180
∴ a + 48 = 180
- Subtract 48 from both sides
∴ a = 132°
∵ 48° is an inscribed angle subtended by the same arc of the
central angle at O (opposite to a)
∵ The measure of the central angle is twice the measure of the
inscribed angle which subtended by the same arc
∴ The measure of the central angle = 2 × 48 = 96°
In the quadrilateral which contains a, 96°, and the other two angles
The sum of the measures of the angles of the quadrilateral is 360°
∴ a + 96 + the sum of the measures of the other 2 angles = 360
∵ a = 132°
∴ 132 + 96 + the sum of the measures of the other 2 angles = 360
- Add the like terms in the left hand side
∴ 228 + the sum of the measures of the other 2 angles = 360
- Subtract 228 from both sides
∴ The sum of the measures of the other 2 angles = 132°
In the inscribed quadrilateral
The sum of the measures of the other 2 angles + a + 20 + b + 48 = 360
∴ 132 + 132 + b + 20 + 48 = 360
- Add the like terms in the left hand side
∴ 332 + b = 360
- Subtract 332 from both sides
∴ b = 28°
